Zeta functions for varieties over finite fields
The Weil conjectures concerns counting solutions to polynomial equations defined over a finite field, and seeing how this number grows as you extend the finite field. This information is best packaged in a function called the zeta function of the corresponding variety. In this video, we introduce this notion and relate it to the usual Riemann zeta function which one finds in number theory. We then mention two parts of Weil's conjecture, the rationality of the zeta function, and the analogue of the functional equation for Riemann's zeta function. In this latter equation, the Euler characteristic of the corresponding variety over the complex numbers makes its appearance.